Error in Byron and Fuller

I've discovered a major logical error in Byron & Fuller's "Mathematics of Classical and Quantum Physics" (Dover edition) and it's really bugging me. It's on pg. 177, in the lead-up to Theorem 4.28. If anybody has read this book and/or knows what I'm talking about, please respond so we can discuss this...I simply can't continue reading in good conscience unless I either resolve the error or find an alternate proof of Thm 4.28